Context. Debris discs are a consequence of the planet formation process and constitute the fingerprints of planetesimal systems. Their solar system counterparts are the asteroid and Edgeworth-Kuiper belts. Aims. The DUNES survey aims at detecting extra-solar analogues to the Edgeworth-Kuiper belt around solar-type stars, putting in this way the solar system into context. The survey allows us to address some questions related to the prevalence and properties of planetesimal systems. Methods. We used Herschel/PACS to observe a sample of nearby FGK stars. Data at 100 and 160 μm were obtained, complemented in some cases with observations at 70 μm, and at 250, 350 and 500 μm using SPIRE. The observing strategy was to integrate as deep as possible at 100 μm to detect the stellar photosphere. Results. Debris discs have been detected at a fractional luminosity level down to several times that of the Edgeworth-Kuiper belt. The incidence rate of discs around the DUNES stars is increased from a rate of ∼12.1% ± 5% before Herschel to ∼20.2% ± 2%. A significant fraction (∼52%) of the discs are resolved, which represents an enormous step ahead from the previously known resolved discs. Some stars are associated with faint far-IR excesses attributed to a new class of cold discs. Although it cannot be excluded that these excesses are produced by coincidental alignment of background galaxies, statistical arguments suggest that at least some of them are true debris discs. Some discs display peculiar SEDs with spectral indexes in the 70-160 μm range steeper than the Rayleigh-Jeans one. An analysis of the debris disc parameters suggests that a decrease might exist of the mean black body radius from the F-type to the K-type stars. In addition, a weak trend is suggested for a correlation of disc sizes and an anticorrelation of disc temperatures with the stellar age.